Is there a way to calculate the closed-form expression of the function that a neural network computes?
To check if a function is linear is easy: if you can train one fully connected layer, without activations, of the right dimensions (for a function $\mathbb{R}^n \rightarrow \mathbb{R}^m$ you need $nm$ weights aka the matrix corresponding to the linear application), with enough data, to 100% accuracy... then it is linear. 

The estimated function is explicit: it is given by the architecture of the NN and its weights. I don't think you can hope for an answer like "$\sin(\pi x + \phi) +4$" if it's not already in the predictive capacity of the NN. Interpretability of NNs is a hot topic. In what cases symbolic reasoning can lead to a simplified expression?